Abstract
In this work, the capacity of multiple-input multiple-output channels that are subject to constraints on the support of the input is studied. The paper consists of two parts. The first part focuses on the general structure of capacity-achieving input distributions. Known results are surveyed and several new results are provided. With regard to the latter, it is shown that the support of a capacity-achieving input distribution is a small set in both a topological and a measure theoretical sense. Moreover, explicit conditions on the channel input space and the channel matrix are found such that the support of a capacity-achieving input distribution is concentrated on the boundary of the input space only. The second part of this paper surveys known bounds on the capacity and provides several novel upper and lower bounds for channels with arbitrary constraints on the support of the channel input symbols. As an immediate practical application, the special case of multiple-input multiple-output channels with amplitude constraints is considered. The bounds are shown to be within a constant gap to the capacity if the channel matrix is invertible and are tight in the high amplitude regime for arbitrary channel matrices. Moreover, in the regime of high amplitudes, it is shown that the capacity scales linearly with the minimum between the number of transmit and receive antennas, similar to the case of average power-constrained inputs.
Highlights
While the capacity of a multiple-input multiple-output (MIMO) channel with an average power constraint is well understood [1], there is surprisingly little known about the capacity of the more practically relevant case in which the channel inputs are subject to amplitude constraints
To the best of our knowledge, the only case for which the input distribution is precisely known is considered in [6], where it is shown for the Gaussian single-input single-output (SISO) channel with an amplitude constraint that two point masses are optimal if amplitude values are smaller than
We studied the capacity of MIMO channels with bounded input spaces
Summary
While the capacity of a multiple-input multiple-output (MIMO) channel with an average power constraint is well understood [1], there is surprisingly little known about the capacity of the more practically relevant case in which the channel inputs are subject to amplitude constraints. Shannon was the first who considered a channel that is constrained in its amplitude [2] In that paper, he derived corresponding upper and lower bounds and showed that in the low-amplitude regime, the capacity behaves as that of channel with an average power constraint. To the best of our knowledge, the only case for which the input distribution is precisely known is considered in [6], where it is shown for the Gaussian SISO channel with an amplitude constraint that two point masses are optimal if amplitude values are smaller than. We make progress on this open problem by deriving several new upper and lower bounds that hold for channels with arbitrary constraints on the support of the channel input distribution and apply them to the practically relevant special case of MIMO channels that are subject to amplitude-constraints
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